In~\cite{reinhardt1998recognizable} K. Reinhard introduced another way to describe
deterministically recognizable languages in 1998. For this he defined a
deterministic process that starts with a given picture $p$ over $\Sigma$ and
ends with the local picture $p'$ over $\Gamma$ using a DS. One step in this
process is the replacement of one symbol $s \in \Sigma$ by its local symbol $g
\in \Gamma$, which can only be performed if it is locally the only possible
choice. Formally this is the following:
\begin{definition}
Let $T = (\Sigma, \Gamma, \Theta, \pi)$ be a domino tiling system. Extend
$\Theta$ to $\Theta' = \Theta \cup \{$\begin{tabular}{|E{0.2cm}|} 
\hline
 s \tabularnewline
\hline
 r \tabularnewline
\hline
\end{tabular}, \begin{tabular}{|E{0.2cm}|} 
\hline
 s \tabularnewline
\hline
 f \tabularnewline
\hline
\end{tabular}, \begin{tabular}{|E{0.2cm}|} 
\hline
 g \tabularnewline
\hline
 r \tabularnewline
\hline
\end{tabular}, \begin{tabular}{|E{0.2cm}|E{0.2cm}|} 
\hline 
q & o \tabularnewline
\hline 
\end{tabular}, \begin{tabular}{|E{0.2cm}|E{0.2cm}|} 
\hline 
q & d \tabularnewline
\hline 
\end{tabular}, \begin{tabular}{|E{0.2cm}|E{0.2cm}|} 
\hline 
e & o \tabularnewline
\hline 
\end{tabular}, $\mid$ \begin{tabular}{|E{0.2cm}|} 
\hline
 g \tabularnewline
\hline
 f \tabularnewline
\hline
\end{tabular}, \begin{tabular}{|E{0.2cm}|E{0.2cm}|} 
\hline 
e & d \tabularnewline
\hline 
\end{tabular} $\in \Theta$, $s = \pi(g), r = \pi(f), q = \pi(e), o = \pi(d)\}$
by also allowing the image symbols in the tiling.

For two intermediate configuations $p, p' \in (\Sigma \cup \Gamma \cup
\{\#\})^{m, n},$ $n, m \geq 1$, we allow a replacement step $p
\underset{T}{\Rightarrow} p'$ if, for all $i \leq m, j \leq n$, we have either
$p(i, j) = p'(i, j)$ or $p(i, j) = \pi(p'(i, j))$ and
\begin{center}
\begin{tabular}{|C{1.6cm}|} 
\hline
 $p(i, j - 1)$ \tabularnewline
\hline
 $p'(i, j)$  \tabularnewline
\hline
\end{tabular}, \begin{tabular}{|C{1.6cm}|} 
\hline
 $p'(i, j)$  \tabularnewline
\hline
 $p(i, j + 1)$ \tabularnewline
\hline
\end{tabular}, \begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline 
 $p(i - 1, j)$ & $p'(i, j)$ \tabularnewline
\hline 
\end{tabular}, \begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline  
 $p'(i, j)$ & $p(i + 1, j)$  \tabularnewline
\hline 
\end{tabular} 
$\in \Theta'$
\end{center} 
and the choice $p'(i,j)$ was 'forced'. Forced means that there is no other $g
\neq p'(i,j)$ in $\Gamma$ with $p(i,j) = \pi(g)$ such that replacing $p(i,j)$ in
$p$ by $g$ would result in each of the $4$ tiles containing this $g$ to be in
$\Theta'$.
\end{definition}
The accepted language for a given DS $T$ is $L_d(T) := \{p \in \Sigma^{*,*} \mid
\hat{p} \overset{*}{\underset{T}{\Rightarrow}} p' \in (\Gamma \cup
\{\#\})^{*,*}\}$. A picture language $L \subseteq \Sigma^{*,*}$ is called
\emph{deterministically recognizable} if there is a DS $T$ with $L = L_d(T)$.
Remark that $L_d(T) \subseteq L(T)$.

The language of pictures over $\{a, b\}$, where all occurring $b$'s
are connected is one example picture language that is deterministically
recognizable. For the proof of this statement see~\cite{reinhardt1998recognizable}.
\label{deterministicprocess}